My knowledge about this family is quite limited. The main reason for describing it anyway is the fact that very little has been written about it until now and perhaps I can help others with the little I know.
The first known person in the family is Hieronymus Boldich (Jeronimus Boldigk). He is mentioned in 1544 (Danske Magazin, 4. row, volume 1, p. 36) when king Christian the third gives him carriage and free ferry to Haderslev. Hieronymus Boldich was an immigrant from the south and he was the secretary of duke Hans the Elder and mayor of Haderslev from appr. 1561-64 until 1572 (probably until his death). Duke Hans himself writes in 1545 about Boldich "who loyally served first the king and thereafter the duke" (Hansborgske Registranter, volume 2, p. 225).
Hieronymus Boldich was married to Margaretha (dead 3 Feb 1582). It is also known that he had a sister who in 1549 was married to Jacob Hansen Guldsmed (Goldsmith). Further it is an established fact - but not widely known - that Hieronymus was the father of the later mayor Christoffer Boldich. The proof is in a number of diplomas in the Arnamagnæan Collection in Copenhagen which is described in detail by Lennart S. Madsen in his article "Kålgårde og Ottinger syd for Haderslev" (cabbage gardens and fields south of Haderslev), Haderslev-Samfundets Årsskrift 1985. Christoffer Boldich was elected mayor in Haderslev in the spring of 1604 and remained in service until 1625-26. he died between 1625 and 1630. He was married twice, 1) with NN who was buried 23 Dec 1593, and 2) with Cirstina who lived in 1636. He had at least five children:
The information above about Christoffer Boldich is from Th.O. Achelis "Haderslev Bys Borgmestre" (The Mayors of Haderslev City), Personalhistorisk Tidsskrift (Danish Genealogical Journal) 1932, pp. 129-145. In an amendment to this article Achelis mentions that Christoffer Boldich in addition had a daughter Anna, married to Gert Busch in Haderslev. In this amendment Achelis refers to an article by Olaus Schmidt in Norsk Slektshistorisk Tidsskrift (Norwegian Genealogical Journal) 1931, p. 8. The same family relation is supposed to be refered to in Hornemann's genealogical collection in Trondheim and is often repeated by Norwegian genealogists.
Another view is presented by the well known Danish genealogist Carl Langholz in "Anetavler for Berømte Danskere" (Ancestry of Famous Danes), 1989. He mentions Anna Boldich as daughter of Hieronymus Boldich and not of Christoffer.
None of the above sources mention any primary sources as evidence of their theories.
Looking at the chronology I tend to favour Carl Langholz' view. It is a weakness that we do not know exactly when these persons were born, but let me give it a try anyway. Langholz thinks that Hieronymus was born appr. 1520. As he is mentioned already in 1544 and has a brother-in-law in 1549, it is reasonable to believ that he was born appr. 1510-20 and had his children in the years between 1540 and 1560. Christoffer Boldich died appr. 1625-30, his first wife died in 1593 and Cirstina survived him. From this information one may assume that he was born appr. 1550. Anna Boldich's and Gert Busch's son Hans Gertsen Busch who became mayor in Trondheim 1630-32 and again in 1635-49 was born in Haderslev either in 1585 or in 1588. This makes it reasonable to believe that Anna Boldich was born around 1560. This chronology has its weaknesses and is not very precise, but it points to Hieronymus - and not Christoffer - being the father of Anna.
Until these relations have been firmly established those who believe Christoffer is the father of Anna can enjoy having another ancestor. As mentioned above it has been established that Hieronymus if the father of Christoffer.
As mentioned Anna was married to Gert Busch, merchant in Haderslev during the 1570s. He was related to (a brother of?) another merchant in Haderslev Marcus (Marx) Busch and they were probably both sons of Hans Busch, mentioned as overseer of the poor (or head of the hospital) in 1554 ("De Hansborgske Domme", p. 301).
Anna Boldich's and Gert Busch's son Hans Gertsen Busch became mayor in Trondheim 1630-32 and again in 1635-49. Before this he was a writer in Skogn in 1620 and he is mentioned as merchant in Trondheim in 1626. He married in 1613 with Maren Svendsdatter, a daughter of the previous mayor in Trondheim Svend Andersen. The couple had three children who survived their father: 1) Giert Hanssøn Busch, a farmer in Sømna, 2) Anne Busch, married to vicar of Skogn Jørgen Pedersen Schjelderup, and 3) Dorothea Busch, married to the vicar in Herøy Hans Anderssen Helkand. In addition Hans Gertsen Busch's tumbstone mentioned that he was buried with the children Anders, Hieronymus, Christopher, Hans, Gertrud and Maren (Svein Tore Dahl, "Embetsmenn i Midt-Norge i tiden 1536-1660" (Officials in Middle Norway 1536-1660), p. 49).
In the book "Slægten Lund m.fl." (The Family Lund and others), Copenhagen 1943, a branch by the name Boldich is mentioned: Rural dean and court vicar in Sønderborg Johan Boldich (1597-1674) married appr. 1631 with Jacobine Heger (Herger?). They had a son, rural dean in Elsinore Ernst Christian Boldich (1647-1706) who married Anna Sofie Stemann (born 1657). They got a son, vicar in Bevtoft Poul Christian Boldich (appr. 1682 - appr. 1726) and a daughter Christine Boldich (1724-1804), married to vicar in Vilstrup Claus Henrik Clausen. I do not know whether this family is related to the one mentioned above.
I hope others can use this information and it goes without saying that I will be happy if anyone can complement it.