use min function with a criteia

If the data is not sorted then

=MIN(IF(A1:A100="name",B1:B100))
committed with CTRL + SHIFT + ENTER

Was the formulae forced into an Array with CTRL + SHIFT + ENTER (if it was it will appear as {=MIN(...)}, if not, reset the Array.

With the data sorted you may be able to avoid an Array, how is the data sorted - by value or by name - and in which order (asc/desc) ?

Yes... CTRL + SHIFT + ENTER

However, if your data is as you say now sorted you can use a non-array approach thanks to the fact that you can use MATCH (with INDEX) to establish the appropriate range to look in (as the data is now effectively grouped together), ie:

=MIN(INDEX(B1:B100,MATCH("name",A1:A100,0)):INDEX(B1:B100,MATCH("name",A1:A100)))

("name" can obviously be a cell containing the name of interest)

Might sound obvious but first point of reference: XL Help... it's useful in understanding the various arguments/parameters used within a Function

In the case of MATCH the final parameter (match_type) is optional (denoted by [ ]) - when omitted it is default of 1, when set to 0 it conducts an exact match based on criteria value so it follows that with data sorted in ASC order:

MATCH(criteria,range,0)

returns first index position of criteria within the range whereas

MATCH(criteria,range)

returns the last instance (parameter omitted means default match_type setting of 1)

I can then use the above with INDEX in reference form to create a range:

INDEX(A:A,MATCH(criteria,range,0)):INDEX(A:A,MATCH(criteria,range))

will say generate a range of A1:A5 (1 & 5 being determined by the respective MATCH functions)

Against which I can apply my *other* function - in this case MIN

=MIN(INDEX(...):INDEX(...))