Group on the person to get the lowest amount for each person, then join the table to get the date for each row:
select y.Amount, y.Date, y.Name
from (
select min(Amount), Name
from TheTable
group by Name
) x
inner join TheTable y on x.Name = y.Name and x.Amount = y.Amount
If the amount can exist on more than one date for a person, pick one of the dates, for example the first:
select y.Amount, min(y.Date), y.Name
from (
select min(Amount), Name
from TheTable
group by Name
) x
inner join TheTable y on x.Name = y.Name and x.Amount = y.Amount
group by y.Amount, y.Name